Package: lmebayesCore 0.1.0

Total variation distances between multivariate normal densities | Definitions | Distance between multivariate normal densities with the same variance-covariance matrix | Distance between multivariate normal densities with the same mean vectors | Distances between multivariate normal densities | Convergence rates for two-block Gibbs samplers for multivariate normal densities | Convergence of mean vectors | Convergence of variance-covariance matrices | Convergence bounds and geometric ergodicity for the two-block Gibbs sampler | Proofs | Proof for densities with shared mean vector | Proof for densities with shared variance-covariance matrix | Proof for convergence of variance-covariance matrices | Proof for convergence of mean vectors | Proof of geometric ergodicity | Special case derivations | How the package evaluates these results | See also

1. Introduction | 1.1 Problem and setting | 1.2 Statement of the main results (informal) | 1.3 Relation to Rosenthal (1995) | 2. The Conditional Bound | 2.1 Two-block Gibbs sampler and the restricted kernel | 2.2 Total variation and L1 distance | 2.3 Truncation: $\pi(\cdot\mid C_d)$ vs. $\pi$ | 2.4 Theorem 1 — geometric convergence of the restricted chain | 3. The Hierarchical GLMM Model | 3.1 Model and hypotheses | 3.2 Propriety of the flat-prior posterior | 3.3 The mean map and the marginal $\gamma$-chain | 3.4 Theorem 2 — existence of a certified safe set (statement) | 4. The Refresh Measure $Q$ | 4.1 Proper population prior ($\Lambda_\gamma\succ0$) | 4.2 Flat-prior limit ($\Lambda_\gamma\downarrow0$) | 4.3 Summary: existence and nondegeneracy of $Q$ | 5. The Minorization Constant and the Certified Set | 5.1 The minorization profile $\varepsilon(\gamma)$ | 5.2 Convexity, coercivity, and compactness of $\widetilde C_d$ | 5.3 The attained constant $\varepsilon_d$ | 5.4 Flat-prior limit ($\Lambda_\gamma\downarrow0$) | 5.5 Closed form under Gaussian closure | 6. Proof of Theorem 2 | 6.1 Assembly of §4–§5 | 6.2 Lifting to the joint $(\gamma,\beta)$ chain | 7. Scope and Extensions | 7.1 Symmetric case: sharper constants | 7.2 What is and is not certified | 7.3 Open problems | Appendix — Proofs | A.1 Proofs for §3.2 (propriety of the flat-prior posterior) and §3.3 (mean map) | A.2 Proofs for §4 | A.3 Proofs for §5 | A.4 Proof of Theorem 2 | References

The theoretical model | Group effects | Population effects | What is fixed and what has a prior | The Big Word Club data | Variables used | The model being estimated | How the population effects are indexed | The priors | See also

Prior specification | Estimating the model using the default settings | Properties of the posterior and the iid sampling procedure | Marginal distribution of the population effects | Conditional distribution of the group effects | Drawing from the posterior | Summarizing the output | Appendix: the reference lmer fit | See also

Prior specification | Estimating the model using the default settings | Properties of the posterior and the two-block Gibbs procedure | Block 1: group effects given the population effects | Block 2: population effects given the group effects | One sweep and one stored draw | Where the sweep count comes from | The rate matrix for this design | From the rate to a certified sweep count | Summarizing the output | Variance convergence during the main phase | See also

Two vignette series, two audiences | The B-series | The model | Symbol table | Internal storage | Full conditionals | Correspondence to lme4 | Model restrictions | Naming conventions in the code | See also

Why two blocks | One sweep | Replicate chains, not one long chain | The call stack | Route selection | Sweep-outer versus chain-outer | Inside Block 1 | Inside Block 2 | Where C++ takes over | What one sweep stores | See also

The scalar picture | The multivariate rate | Computing it without forming (P_{22}) | A worked case with a closed form | From the rate to a total-variation bound | The full-rank conditions | Level 1: each group's design must be full column rank | Level 2: the retained groups must identify (\gamma) | Beyond the Gaussian case | See also

The pipeline | What tv_tol means | Why one extra sweep | Starting points and the (D_0) term | The pilot / main cost trade-off | From a pfamily_list | When certification fails | See also

Chapter B04: The known-vcov Gaussian route | Route selection | Why the route is exact | Two engines behind one route | The exact iid engine | The symmetry precondition | The Gibbs engine on an exact target | The precision matrix P | The compiled kernel | Working example | When the route does not apply | See also

Chapter B05: Estimated variance components | Block 2 as a regression on the random effects | Two ways to update | Method A: closed-form conditional mode | Method B: rglmb(), the production path | Plug-ins and where the prior is calibrated | The truncation window | What the chain looks like | Why a pilot stage becomes necessary | Selecting components individually | See also

Why a single (\lambda^*) stops being enough | What the engine certifies instead | The extended rate | The new blocks | The residuals only make it worse | Using it | Reading the output | Two checks worth running | Scanning over draws | The marginal route | Theorem or heuristic: the honest table | See also

The model, and what dispformula selects | Calibrating (\sigma^2_j): why the within-group RSS is the wrong target | The marginal sum of squares | The Gamma parameters | rate versus rate_gamma | Folding in hyperparameter uncertainty | The truncation window | dGamma_list() | What Block 1 does with it | Effect on the rate | Caveats | See also

What model_setup() returns | Level 1: within-group rank | What failure means, and what it does not | The estimability check | Level 2: across-group rank | Why this one is fatal | Fixed-effect grammar | Reading the print output | A worked failure | See also

The engines | The partition object | Block 1 of the mixed-model sampler | lmbBlock() and glmbBlock() are not these engines | Per-block prior calibration | Identifiability, without a second level | Who owns what | See also

Why the ensemble view is the right one | The object | Which engines populate it | Reading the two plots | plot_mean_convergence() | plot_var_convergence() | Multiple chains and n_chains | Using the history programmatically | Interpreting a pilot history | See also